The key theorem you need from elementary topology is that any surjective map from a compact space to a Hausdorff space is a quotient map. To apply that theorem you must set up the appropriate quotient maps. This is a purely topological problem, requiring you to guess the correct formulas for the appropriate quotient maps.
You have described constructions 1 and 2, but let me also describe a construction which which is closest to the definition of $\mathbb{C}P^n$:
- Construction 0: $\mathbb{C}P^n$ is the quotient of $\mathbb{C}^{n+1}-\{0\}$ defined by the equivalence relation $x \sim zx$ for each $x \in \mathbb{C}^{n+1}-\{0\}$ and each $\lambda \in \mathbb{C} - \{0\}$.
Let $f : \mathbb{C}^{n+1}-\{0\} \to \mathbb{C}P^n$ be the function such that $f(x)$ equals the equivalence class of $x$. The topology on $\mathbb{C}P^n$ is defined to be the unique one such that $f$ is a quotient map.
Next let me compare construction 0 and construction 1. Let $g : S^{2n+1} \to \mathbb{C}P^n$ be the restriction of the function $f$ to the unit sphere $S^{2n+1} \in \mathbb{C}^{n+1}-\{0\} = \mathbb{R}^{2n+2}-\{0\}$. According to the definition of $f$, two points $x,y \in S^{2n+1}$ satisfy $g(x)=g(y)$ if and only if there exists $\lambda \in \mathbb{C}-\{0\}$ such that $x=\lambda y$ (notice that $\lambda$ must also have norm $1$ since $x$ and $y$ have norm $1$). Since the domain of $g$ is compact and its range is Hausdorff, the above theorem applies, and therefore $g$ is a quotient map. It follows that $\mathbb{C}P^n$ is homeomorphic to the quotient space obtained from $S^{2n+1}$ by identifying $x,y$ if and only if there exists $\lambda \ne 0$ such that $x=\lambda y$ (again the $\lambda$ must have norm $1$).
Now, to get to the heart of your question, let's compare construction 1 and construction 2. The method is similar: construct a quotient map $h : \mathbb{D}^{2n} \to \mathbb{C}P^n$ such that $h(x)=h(y)$ if and only if $x,y \in \partial\mathbb{D}^{2n}$ and $x=\lambda y$ for a nonzero complex constant (which, again, must have norm $1$). We will construct $h$ using $g$. To do this, consider the inclusion $\mathbb{D}^{2n} \subset \mathbb{C}^{n} \subset \mathbb{C}^{n+1}$. Map $\mathbb{D}^{2n}$ to $S^{2n+1}$ using the function $$p(a_1,b_1,...,a_n,b_n,0,0) = (a_1,b_1,...,a_n,b_n,\,\sqrt{1 - (a_1^2+b_1^2+...+a_n^2+b_n^2)}\,\,,\,\,0)
$$
We then have a map
$$h = g \circ p : \mathbb{D}^{2n} \to \mathbb{C}P^n
$$
Now check that this map $h$ is surjective, its domain is compact, and its range is Hausdorff. Applying the key theorem, $h$ is a quotient map. Also check that $h(x)=h(y)$ if and only if $x=y$ or $x,y \in \partial \mathbb{D}^{2n}$ and $x = \lambda y$ for some $\lambda \in \mathbb{C}$ (which as said must have norm $1$). It follows that $\mathbb{C}P^n$ has the description in your construction 2.
